On graded quasi-semiprime submodules of graded modules over graded commutative rings (original) (raw)

Let G be a group with identity e. Let R be a Ggraded commutative ring and M a graded R-module. A proper graded submodule N of M is called a graded semiprime submodule if whenever r ∈ h(R), m ∈ h(M) and n ∈ Z with rm ∈ N , then rm ∈ N . In this paper, we introduce the concept of graded quasisemiprime submodule as a generalization of graded semiprime submodule and show a number of results in this class. We say that a proper graded submodule N of M is a graded quasi-semiprime submodule if (N :R M) is a graded semiprime ideal of R.